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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">d1</font> being    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font> ex <font color="Olive" title="b2">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font>,<font color="Maroon" title="c3">v</font>) );<br/>
<a NAME="E1:103_1_1"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="103_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:103_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">d1</font>, <font color="Maroon" title="c6">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:103_1_1_1"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
( <font color="Maroon" title="c5">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) )
 <span class="kw">and </span><br/><a NAME="E4:103_1_1_1"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">d1</font> = <font color="Maroon" title="c5">d1</font> as    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103_1_1_1"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="nomin_1.html#T43" target="_self" onmouseover="rs('nomin_1/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<span class="kw">take </span>
 <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">d1</font>,<font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c3">v</font>)
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">d1</font>,<font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c3">v</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>, <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">d1</font>,<font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c3">v</font>)] )</span><br/>

<span class="kw">thus </span><a NAME="E6:103_1_1_1"/>
(  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">d1</font>,<font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c3">v</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>, <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">d1</font>,<font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c3">v</font>)] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:103_1_1_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:103_1_1"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:103_1_1"/><span class="lab"><font color="Green" title="E57">A5</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)
 <span class="kw">and </span><br/><a NAME="E5:103_1_1"/><span class="lab"><font color="Green" title="E58">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S6" onmouseover="rs('funct_1/S6')" onmouseout="rh()">FUNCT_1:sch 6</a>(<a class="txt" href="#E1:103_1_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103_1_1"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="txt" href="#E4:103_1_1"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a>, <a class="ref" href="zfmisc_1.html#T96" onmouseover="rs('zfmisc_1/T96')" onmouseout="rh()">ZFMISC_1:96</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">f</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; (  for <font color="Olive" title="b1">d1</font> being    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font><br/>  for <font color="Olive" title="b2">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font>,<font color="Maroon" title="c3">v</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:103_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103_1_1"><span class="lab"><font color="Green" title="E56">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d1</font> being    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font><br/>  for <font color="Olive" title="b2">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font>,<font color="Maroon" title="c3">v</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">D1</font> be    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Maroon" title="c6">D1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">D1</font>,<font color="Olive" title="b1">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">D1</font>,<font color="Olive" title="b1">d2</font>,<font color="Maroon" title="c3">v</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c7">D2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">D1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font>,<font color="Maroon" title="c3">v</font>) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c8">D</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:103_1_1"/>
(  not <font color="Maroon" title="c6">D1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font>,<font color="Maroon" title="c3">v</font>)</span><br/>

<a NAME="E9:103_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">D1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nomin_1.html#T42" target="_self" onmouseover="rs('nomin_1/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E10:103_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103_1_1"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">d1</font> being    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c10">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E12:103_1_1"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">d1</font>,<font color="Maroon" title="c10">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E13:103_1_1"/><span class="lab"><font color="Green" title="E60">A8</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c9">d1</font>,<font color="Maroon" title="c10">d2</font>,<font color="Maroon" title="c3">v</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:103_1_1"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="txt" href="#E5:103_1_1"><span class="lab"><font color="Green" title="E58">A6</font></span></a></span>;<br/>
<a NAME="E14:103_1_1"/>
( <font color="Maroon" title="c9">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D1</font> &amp; <font color="Maroon" title="c10">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:103_1_1"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:103_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">D1</font>,<font color="Maroon" title="c7">D2</font>,<font color="Maroon" title="c3">v</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:103_1_1"><span class="lab"><font color="Green" title="E60">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
